Synchronous Machine Models

This page describes the most common synchronous machine models used in stability studies.

Nomenclature

The standard machine parameters are defined as follows:

• $R_{a}\,$ is the armature resistance (pu)
• $X_{a}\,$ is the armature reactance (pu)
• $X_{d}\,$ is the d-axis synchronous reactance (pu)
• $X_{q}\,$ is the q-axis synchronous reactance (pu)
• $X'_{d}\,$ is the d-axis transient reactance (pu)
• $X'_{q}\,$ is the q-axis transient reactance (pu)
• $X''_{d}\,$ is the d-axis subtransient reactance (pu)
• $X''_{q}\,$ is the q-axis subtransient reactance (pu)
• $T'_{d0}\,$ is the d-axis transient open loop time constant (s)
• $T'_{q0}\,$ is the q-axis transient open loop time constant (s)
• $T''_{d0}\,$ is the d-axis subtransient open loop time constant (s)
• $T''_{q0}\,$ is the q-axis subtransient open loop time constant (s)
• $H\,$ is the machine inertia constant (MWs/MVA)
• $D\,$ is an additional damping constant (pu)

Note that per-unit values are usually expressed on the machine's MVA base.

6th Order (Sauer-Pai) Model

6th order synchronous machine model based on the book:

Sauer, P.W., Pai, M. A., "Power System Dynamics and Stability", Stipes Publishing, 2006

Stator magnetic equations:

${\dot {E'_{q}}}={\frac {1}{T'_{d0}}}\left[V_{fd}-E'_{q}-(X_{d}-X'_{d})\left(I_{d}-\gamma _{d2}\psi ''_{d}-(1-\gamma _{d1})I_{d}+\gamma _{d2}E'_{q}\right)\right]\,$

${\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[-E'_{q}-(X_{q}-X'_{q})\left(I_{q}-\gamma _{q2}\psi ''_{q}-(1-\gamma _{q1})I_{q}-\gamma _{q2}E'_{d}\right)\right]\,$

${\dot {\psi ''_{d}}}={\frac {1}{T''_{d0}}}\left[E'_{q}-\psi ''_{d}-(X'_{d}-X_{a})I_{d}\right]\,$

${\dot {\psi ''_{q}}}={\frac {1}{T''_{q0}}}\left[-E'_{d}-\psi ''_{q}-(X'_{q}-X_{a})I_{q}\right]\,$

$\psi _{d}=-X''_{d}I_{d}+\gamma _{d1}E'_{q}+(1-\gamma _{d1})\psi ''_{d}\,$

$\psi _{q}=-X''_{q}I_{q}-\gamma _{q1}E'_{d}+(1-\gamma _{q1})\psi ''_{q}\,$

where $\gamma _{d1}={\frac {X''_{d}-X_{a}}{X'_{d}-X_{a}}}\,$

$\gamma _{q1}={\frac {X''_{q}-X_{a}}{X'_{q}-X_{a}}}\,$

$\gamma _{d2}={\frac {1-\gamma _{d1}}{X'_{d}-X_{a}}}\,$

$\gamma _{q2}={\frac {1-\gamma _{q1}}{X'_{q}-X_{a}}}\,$

Stator electrical equations (neglecting electromagnetic transients):

$V_{d}=-\omega \psi _{q}-R_{a}I_{d}\,$

$V_{q}=\omega \psi _{d}-R_{a}I_{q}\,$

Equations of motion:

${\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,$

${\dot {\delta }}=\Omega _{s}(\omega -\omega _{s})\,$

Initialisation:

${\boldsymbol {E}}_{q0}={\boldsymbol {V}}_{t0}+(R_{a}+jX_{q})\times {\boldsymbol {I}}_{a0}\,$

$\delta _{0}=\angle {\boldsymbol {E}}_{q0}\,$

$\psi _{0}=\angle {\boldsymbol {I}}_{a0}\,$

$I_{d0}=|{\boldsymbol {I}}_{a0}|\sin(\delta _{0}-\psi _{0})\,$

$I_{q0}=|{\boldsymbol {I}}_{a0}|\cos(\delta _{0}-\psi _{0})\,$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{d0}=|{\boldsymbol {V}}_{t0}|\sin(\delta _{0}-\theta _{0})\,}

$V_{q0}=|{\boldsymbol {V}}_{t0}|\cos(\delta _{0}-\theta _{0})\,$

$E'_{d0}=V_{d}-X''_{q}I_{q0}+R_{a}I_{d0}-(1-\gamma _{q1})(X'_{q}-X_{a})I_{q0}\,$

$E'_{q0}=V_{q}+X''_{d}I_{d0}+R_{a}I_{q0}+(1-\gamma _{d1})(X'_{d}-X_{a})I_{d0}\,$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \psi ''_{d0}=E'_{q0}-(X'_{d}-X_{a})I_{d0}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \psi ''_{q0}=-E'_{d0}-(X'_{q}-X_{a})I_{q0}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{fd0}=E'_{q0}+(X_{d}-X'_{d})\left(I_{d0}-\gamma _{d2}\psi ''_{d0}-(1-\gamma _{d1})I_{d0}+\gamma _{d2}E'_{q0}\right)\,}

$P_{m}=P_{e0}=(V_{d0}+R_{a}I_{d0})I_{d0}+(V_{q0}+R_{a}I_{q0})I_{q0}\,$

$\omega _{0}=\omega _{s}\,$

6th Order (Anderson-Fouad) Model

6th order synchronous machine model based on the book:

Anderson, P. M., Fouad, A. A., "Power System Control and Stability", Wiley-IEEE Press, New York, 2002

Stator magnetic equations:

${\dot {E'_{q}}}={\frac {1}{T'_{d0}}}\left[V_{fd}-(X_{d}-X'_{d})I_{d}-E'_{q}\right]\,$

${\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[(X_{q}-X'_{q})I_{q}-E'_{d}\right]\,$

${\dot {E''_{q}}}={\frac {1}{T''_{d0}}}\left[E'_{q}-(X'_{d}-X''_{d})-E''_{q}\right]\,$

${\dot {E''_{d}}}={\frac {1}{T''_{q0}}}\left[E'_{d}-(X'_{q}-X''_{q})-E''_{d}\right]\,$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E''_{q}- V_{q} = R_{a} I_{q} + X''_{d} I_{d} \, }

$E''_{d}-V_{d}=R_{a}I_{d}-X''_{q}I_{q}\,$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{d} = E''_{q} - X''_{d} I_{d} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{q} = -E''_{d} - X''_{q} I_{q} \,}

Stator electrical equations (neglecting electromagnetic transients):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{d} = -\omega \psi_{q} - R_{a} I_{d} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{q} = \omega \psi_{d} - R_{a} I_{q} \,}

Equations of motion:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\omega} = \frac{1}{2H} \left[ P_{m} - P_{e} - D(\omega - \omega_{s}) \right] \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\delta} = \Omega_{s} (\omega - \omega_{s}) \, }

Initialisation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X_q) \times \boldsymbol{I}_{a0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_0 = \angle \boldsymbol{E}_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_0 = \angle \boldsymbol{I}_{a0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{d0} = |\boldsymbol{I}_{a0}| \sin (\delta_0 - \phi_0 ) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{q0} = |\boldsymbol{I}_{a0}| \cos (\delta_0 - \phi_0 ) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{fd0} = |\boldsymbol{E}_{q0}| + (X_{d} - X_{q}) I_{d0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{q0} = V_{fd0} - (X_{d} - X'_{d}) I_{d0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E''_{q0} = E'_{q0} - (X'_{d} - X''_{d}) I_{d0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{d0} = (X_{q} - X'_{q}) I_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E''_{d0} = E'_{d0} + (X'_{q} - X''_{q}) I_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{d0} = E''_{d0} + X''_{q} I_{q0} - R_{a} I_{d0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{q0} = E''_{q0} - X''_{d} I_{d0} - R_{a} I_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_m = P_{e0} = (V_{d0} + R_{a} I_{d0}) I_{d0} + (V_{q0} + R_{a} I_{q0}) I_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \omega_s \,}

4th Order (Two-Axis) Model

Stator magnetic equations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{E'_{q}} = \frac{1}{T'_{d0}} \left[ V_{fd} - (X_{d} - X'_{d})I_{d} - E'_{q} \right] \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{E'_{d}} = \frac{1}{T'_{q0}} \left[ (X_{q} - X'_{q})I_{q} - E'_{d} \right] \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{q}- V_{q} = R_{a} I_{q} + X'_{d} I_{d} \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{d}- V_{d} = R_{a} I_{d} - X'_{q} I_{q} \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{d} = E'_{q} - X'_{d} I_{d} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{q} = -E'_{d} - X'_{q} I_{q} \,}

Stator electrical equations (neglecting electromagnetic transients):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{d} = -\omega \psi_{q} - R_{a} I_{d} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{q} = \omega \psi_{d} - R_{a} I_{q} \,}

Equations of motion:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\omega} = \frac{1}{2H} \left[ P_{m} - P_{e} - D(\omega - \omega_{s}) \right] \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\delta} = \Omega_{s} (\omega - \omega_{s}) \, }

Initialisation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X_q) \times \boldsymbol{I}_{a0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_0 = \angle \boldsymbol{E}_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_0 = \angle \boldsymbol{I}_{a0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_0 = \angle \boldsymbol{V}_{t0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{d0} = |\boldsymbol{I}_{a0}| \sin (\delta_0 - \phi_0 ) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{q0} = |\boldsymbol{I}_{a0}| \cos (\delta_0 - \phi_0 ) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{d0} = |\boldsymbol{V}_{t0}| \sin (\delta_0 - \theta_0 ) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{q0} = |\boldsymbol{V}_{t0}| \cos (\delta_0 - \theta_0 ) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{q0} = V_{q0} + R_a I_{q0} + X'_{d} I_{d0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{d0} = V_{d0} + R_a I_{d0} - X'_{q} I_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{fd0} = E'_{q0} + (X_d - X'_d) I_{d0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_m = P_{e0} = (V_{d0} + R_{a} I_{d0}) I_{d0} + (V_{q0} + R_{a} I_{q0}) I_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \omega_s \,}

2nd Order (Classical) Model

Stator equations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'_{q} - V_{q} = R_{a} I_{q} + X'_{d} I_{d} \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{d} = X'_{d} I_{q} - R_{a} I_{d} \, }

Equations of motion:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\omega} = \frac{1}{2H} \left[ P_{m} - P_{e} - D(\omega - \omega_{s}) \right] \, }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\delta} = \Omega_{s} (\omega - \omega_{s}) \, }

Initialisation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X'_d) \times \boldsymbol{I}_{a0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_0 = \angle \boldsymbol{E}_{q0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_0 = \angle \boldsymbol{V}_{t0} \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_m = P_{e0} = \left( \frac{1}{R_a + j X'_d} \right) |\boldsymbol{V}_{t0}| |\boldsymbol{E}_{q0}| \sin(\delta_0 - \theta_0) \,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \omega_s \,}